Authors

Abstract

Hidden Markov Models, HMM's, are mathematical models of Markov processes with state that is hidden, but from which information can leak. They are typically represented as 3-way joint-probability distributions.
We use HMM's as denotations of probabilistic hidden-state sequential programs: for that, we recast them as "abstract" HMM's, computations in the Giry monad $\Dist$, and equip we them with a partial order of increasing security. However to encode the monadic type with hiding over some state X we use DX->D^2X rather than the conventional X->DX that suffices for Markov models whose state is not hidden. We illustrate the DX->D^2X construction with a small Haskell prototype.
We then present uncertainty measures as a generalisation of the extant diversity of probabilistic entropies, with characteristic analytic properties for them, and show how the new entropies interact with the order of increasing security. Furthermore, we give a "backwards" uncertainty-transformer semantics for HMM's that is dual to the "forwards" abstract HMM's --- it is an analogue of the duality between forwards, relational semantics and backwards, predicate-transformer semantics for imperative programs with demonic choice.
Finally, we argue that, from this new denotational-semantic viewpoint, one can see that the Dalenius desideratum for statistical databases is actually an issue in compositionality. We propose a means for taking it into account.